In the text's rules, no. I drift it sometimes and just do it openly, most often because I'm teaching the system. In the non-demo or non-con play, I tend to keep it secret.

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How exactly can binding rolls be re-rolled during play?

Um ... you mean, if a sorcerer has bound a demon, and then binds it again? That's not really something that happens; if the demon is currently bound, then that's that. If the sorcerer wants to make the binding stronger (I suppose in his or her favor), then he or she simply needs to favor the demon's Need and/or Desire.

Is there a specific text passage that leads to this question? Because I can think of some circumstances in which "re-binding" might be involved, but they need some context.

Peter, what exactly are you asking? "What's the benefit" sounds specific, but for this topic, and for Sorcerer, it's not.

I have an idea - to make this discussion work, try this: state a benefit that you think the secrecy might provide, to some groups and in some way, and then state why keeping the binding strength be known is better for you.

Really, this will help us actually communicate instead of vaguely posting about benefits in some unknown play-context with unknown participants.